Quantum Phase Estimation

Quantum Phase Estimation (QPE) is a quantum algorithm that estimates the phase (eigenvalue) associated with an eigenvector of a unitary operator. It is fundamental in various quantum algorithms, including Shor's algorithm for integer factorization.

• **Eigenvalue**: The value that describes how the eigenvector is scaled during a linear transformation.
• **Unitary Operator**: An operator that preserves the inner product and thus the length of quantum states.
• **Quantum Register**: A collection of qubits used to store quantum information.

The QPE algorithm consists of several key steps:

1. Prepare the quantum state.
2. Apply a series of controlled unitary operations.
3. Perform inverse Quantum Fourier Transform (QFT).
4. Measure the qubits to obtain the phase estimate.

The following is a simple implementation of QPE using Qiskit:

from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_histogram

def qpe_amod15(a):
    n_count = 4  # Number of counting qubits
    qc = QuantumCircuit(n_count + 4, n_count)

    # Initialize counting qubits
    for q in range(n_count):
        qc.h(q)  # Apply Hadamard gate to each counting qubit

    # Apply controlled unitary operations
    for q in range(n_count):
        qc.append(c_amod15(a, 2**q), 
                  [q] + [i + n_count for i in range(4)])

    # Apply inverse QFT
    qc.append(qft(n_count, inverse=True), range(n_count))

    # Measure the counting qubits
    qc.measure(range(n_count), range(n_count))

    return qc

# Simulate the circuit
backend = Aer.get_backend('qasm_simulator')
qc = qpe_amod15(7)
results = execute(qc, backend).result()
counts = results.get_counts(qc)
plot_histogram(counts)
                

When implementing QPE, consider the following:

• Ensure qubits are properly initialized and entangled.
• Optimize the number of qubits for the desired precision.
• Account for noise in quantum circuits, especially in real hardware.